Numbers and Sets

Equations and Inequalities (restricted to first and second order or traceable to first and second order)

Basic functions

Tools for the applications (Straight line equation, sequences, sequence limits, function limits, derivative and their algebra)

Applications to the differential calculus (polinomial and rational functions' graphs, linar approximations)

Integral calculus (Theory and calculus chief methods)

Differential equations (Theory and calculus of the linear first order equations)

Elements of descriptive statistics

Problems of descriptive and inferential statistics; discrete and continuous variables; character; sample; absolute frequency; relative frequency; statistical variable; point diagram; bar chart; histograms; aerograms; Box plots; statistics; fashion; median; cumulated frequencies and relative histogram; quartiles; quantile; arithmetic average; geometric mean; harmonic mean; mean square deviation and deviation; data range; deviance; variance and calculation formula; standard deviation.

Elements of probability calculation

Combinatorial calculus: counting principle, simple permutations and theorem, permutations with repetition and theorem, simple combinations and theorem, binomial coefficient, simple dispositions and theorem, dispositions with repetitions and theorem.

The case: historical references; Events and their algebra: standard event, implication between events, equal events, opposite event, logical sum, logical product, property of events, incompatible events, logically independent events, sample space, conditioned events.

The different definitions of probability: classical, frequentistic, axiomatic, subjective, elementary propositions on the probability of events. Probability assessments in the hypothesis of equally probable elementary cases: card games, birthday problem.

Notes on conditional probability: compound probability theorem, stochastically independent events, Bayes formula, Bayes theorem, a priori probability, posterior probability. Applications

Insights into probability and statistics

Random variable, expected value of a random variable, rejection variable, variance and standard deviation of a random variable.

Bienaymé-Tchebyceff torema.

Proposition and corollary of the alternative formula for calculating variance. Uniform probability distribution, distribution function of a random variable, property of the distribution function.

Operations with random variables: joint probability and marginal probability, Cantelli's theorem, independent random variables, proposition of the expected value of the product of two random variables.

The regression line: covariance of two random variables, proposition of the alternative formula for calculating covariance, proposition and corollary of the variance of the sum of two random variables, correlation coefficient.

The binomial random variable: the scheme of repeated or Bernoulli trials, the binomial distribution, proposition of the expected value and variance of the binomial variable, proposition of the expected value and variance of the relative frequency of success, the weak law of large numbers.

Numbers late for the lotto game.

The random Poisson variable: the generalized Poisson scheme, the generalized Poisson distribution, proposition expected value and variance of the generalized Poisson variable. Poisson's scheme, Poisson's distribution, relationship between Poisson's law and binomial law, proposition of the expected value and variance of the Poisson variable.

The hypergeometric random variable: the hypergeometric scheme of a succession of dependent tests, the hypergeometric distribution, relationship between the hypergeometric law and the binomial law, proposition of the expected value and variance of the hypergeometric variable.